Long Term Evolution (LTE) corresponds to the more recent development in wireless digital communications, following the High Speed Downlink Packet Access HSDPA and High Speed Uplink Packet Access (HSUPA).
Synchronization in LTE communication uses two distinctive synchronization signals, i.e. a so-called Primary Synchronization Signal (PSS) and the so-called Secondary Synchronization Signal (SSS).
The PSS signal is a signal which is known to the UE and which is periodically sent by the cell or basestation and received by the User Equipment (UE) for detecting the initial timing and for computing the strength of the signal.
Once the PSS signal is detected, the UE proceeds with the detection of the SSS which allows completion of the synchronization process and detect the start of the frame as well as the identification of the cell.
Generally speaking, the SSS signal is a 62-bit length sequence based on the use of two interleaved sequences (each having a length of 31 bits) being each selected from a set of 31 known sequences.
The sequence S(0), . . . , S(61) used for the second synchronization signal is an interleaved concatenation of two length-31 binary sequences. The concatenated sequence is scrambled with a scrambling sequence given by the primary synchronization signal.
Let Sm=[sm[1], sm[2], . . . , sm[N]]T be the frequency domain length-62 transmitted SSS sequence (not including DC).
One may write the received signal on the central 62 sub-carriers as follows,Y=SmH+νwhere Y is the N×1 received signal vector, H=[H1, H2, . . . , HN]T is the channel gain vector and ν is the noise vector. Sm is a matrix defined in function of the transmitted sequence by.Sm=diag(sm[1], sm[2], . . . , sm[N])
One considers Maximum Likelihood detection of the SSS sequence and analysis both coherent and non coherent detections.
Two basic techniques are basically known for achieving detection of the SSS signal: 1. the non coherent method and 2. the coherent method.
1. First Known Technique: Non Coherent Detection of the SSS
Non coherent detection assumes that the channel gains are not known to the detector. The Maximum Likelihood detection of the SSS sequence consists on finding the sequence maximizing the probability of receiving Y knowing that the sequence Sm is transmittedp(Y|Sm)The signal vector Y is Gaussian conditionally on the transmitted sequence Sm and on the channel gains vector H. Its conditional probability (with the assumption of white i.i.d. Gaussian noise with variance N0 i.e. Rn=N0I) is given by
      p    ⁡          (                        Y          ❘                      S            m                          ,        H            )        =            1                        (                      π            ⁢                                                  ⁢                          N              0                                )                N              ⁢          exp      ⁡              (                  -                                                                      (                                      Y                    -                                                                                            S                          _                                                m                                            ⁢                      H                                                        )                                N                            ⁢                              (                                  Y                  -                                                                                    S                        _                                            m                                        ⁢                    H                                                  )                                                    N              0                                      )            p(Y|Sm) can be obtained from p(Y|Sm, H) by averaging with respect to the channel vector H.
      p    ⁡          (              Y        ❘                  S          m                    )        =            ∫      H        ⁢                  p        ⁡                  (                                    Y              ❘                              S                m                                      ,            H                    )                    ⁢              p        ⁡                  (          H          )                    ⁢              ⅆ        H            where
      p    ⁡          (      H      )        =            1                        π          N                ⁢                  det          ⁡                      (                                          R                _                            H                        )                                ⁢          exp      ⁡              (                              -                          H              H                                ⁢                                    R              _                        H                          -              1                                ⁢          H                )            Thus we have
            S      ^        m    =      arg    ⁢                  ⁢    max    ⁢          {                                                  1                                                                    (                                          π                      ⁢                                                                                          ⁢                                              N                        0                                                              )                                    N                                ⁢                                  π                  N                                ⁢                                  det                  ⁡                                      (                                                                  R                        _                                            H                                        )                                                                                                                                          ∫                H                            ⁢                                                exp                  (                                      -                                          (                                                                                                                                                                  (                                                                  Y                                  -                                                                                                                                                    S                                        _                                                                            m                                                                        ⁢                                    H                                                                                                  )                                                            H                                                        ⁢                                                          (                                                              Y                                -                                                                                                                                            S                                      _                                                                        m                                                                    ⁢                                  H                                                                                            )                                                                                                            N                            0                                                                          +                                                                              H                            H                                                    ⁢                                                                                    R                              _                                                        H                                                          -                              1                                                                                ⁢                          H                                                                    )                                                        )                                ⁢                                  ⅆ                  H                                                                        }      After some computations and simplifications we find that
            ∫      H        ⁢                  exp        ⁡                  (                      -                          (                                                                                                                  (                                                  Y                          -                                                                                                                    S                                _                                                            m                                                        ⁢                            H                                                                          )                                            H                                        ⁢                                          (                                              Y                        -                                                                                                            S                              _                                                        m                                                    ⁢                          H                                                                    )                                                                            N                    0                                                  +                                                      H                    H                                    ⁢                                                            R                      _                                        H                                          -                      1                                                        ⁢                  H                                            )                                )                    ⁢              ⅆ        H              =                    π        N                    det        ⁡                  (                      A            _                    )                      ⁢          exp      ⁡              (                                            B              m              H                        ⁢                                          A                _                            m                              -                1                                      ⁢                          B              m                                -          C                )            Where
                    A        _            m        =                            1                      N            0                          ⁢                              S            _                    m          H                ⁢                              S            _                    m                    +                        R          _                H                  -          1                                B      m        =                  1                  N          0                    ⁢                        S          _                m        H            ⁢      Y      ⁢                          ⁢      and            C    =                  1                  N          0                    ⁢              Y        H            ⁢      Y      Since Sm is a diagonal matrix with binary elements, we have that SmHSm=I. Hence we have
            A      _        m    =                    1                  N          0                    ⁢      I        +                  R        _            H              -        1            which does not dependent on m.Since also the value of C does not depend on m, the detection decision reduces toŜm=arg max{exp(BmHAm−1Bm)}The logarithm function is a monotonic, thus the detected sequence is the one maximizingBmHAm−1Bm 
The non coherent detection shows some significant drawbacks.
Firstly, the non-coherent detection performances strongly depend on the channel coherence bandwidth. The non-coherent detection method work very well when the channel is frequency flat, but this is not the case in reality and thus the SSS decoding performance will degrade and are not robust to the different propagation environment.
Secondly, the non-coherent detection significantly depends on the error on timing. In fact a timing offset will introduce a phase rotation in the frequency domain which will make the non-coherent detection impractical.
2. Second Known Technique: Coherent Detection of the SSS
The second known technique is the coherent technique which assumes the knowledge of the channel characteristics (ie the frequency response the channel) what may results, for instance, from the already achieved detection of the PSS which, when used as a pilot, may also serve for channel estimation.
Under this assumption, the Maximum Likelihood detection of the SSS sequence consists on finding the sequence maximizing the probability of receiving Y knowing that the sequence Sm is transmitted.
The conditional pdf of Y given Sm and His given by
                              p          ⁡                      (                                          Y                ❘                                  S                  m                                            ,              H                        )                          =                              1                                          (                                  π                  ⁢                                                                          ⁢                                      N                    0                                                  )                            N                                ⁢                      exp            (                          -                                                                                          (                                              Y                        -                                                                                                            S                              _                                                        m                                                    ⁢                          H                                                                    )                                        H                                    ⁢                                      (                                          Y                      -                                                                                                    S                            _                                                    m                                                ⁢                        H                                                              )                                                                    N                  0                                                      )                                                  =                              1                                          (                                  π                  ⁢                                                                          ⁢                                      N                    0                                                  )                            N                                ⁢                      exp            (                          -                                                                                                              Y                      -                                                                                                    S                            _                                                    m                                                ⁢                        H                                                                                                  2                                                  N                  0                                                      )                              The ML decision rule is then,
                                          S            ^                    m                =                  arg          ⁢                                          ⁢          min          ⁢                      {                                                                            Y                  -                                                                                    S                        _                                            m                                        ⁢                    H                                                                              2                        }                                                  =                  arg          ⁢                                          ⁢          min          ⁢                      {                                          ∑                                  i                  =                  0                                                  N                  -                  1                                            ⁢                                                                                                            Y                      ⁡                                              [                        i                        ]                                                              -                                                                  H                        i                                            ⁢                                                                        S                          m                                                ⁡                                                  [                          i                          ]                                                                                                                                      2                                      }                              
When using the channel estimation from the PSS, a problem may arise in the case where another base station, time-synchronized with the target base station, has the same PSS. In this case the estimated channel will be the sum of the channel from the targeted cell and the interfering cell with the same PSS. This error on the channel estimation will result in a considerable degradation of performance in the case of coherent detection.
This is a clear drawback of the coherent detection method and there is therefore a need for an alternative method for detecting the SSS which overcomes those drawbacks.